Exterior Algebra and Differential Forms

Differential $p$-forms are smooth alternating covariant tensors. Their alternation makes a form evaluate oriented $p$-dimensional parallelepipeds, while the wedge product assembles lower-degree measurements into higher-dimensional ones. These properties, together with the determinant transformation of top forms, make forms the natural objects to integrate without choosing a metric.

Required background. Tensor Fields and Natural Operations — use covariant tensors, pullbacks, tensor products, and weighted antisymmetrization.

Helpful background. Calculus and Linear Algebra for Relativity — use determinants, alternating multilinear maps, and multivariable coordinate changes.

Alternating tensors and form degree

For a vector space $V$, an alternating $p$-tensor $\omega\in\Lambda^pV^*$ changes sign when two arguments are exchanged and vanishes if two arguments coincide. On an $n$-manifold, a differential $p$-form is a smooth section of $\Lambda^pT^*M$. In coordinates,

$$\omega=\frac{1}{p!}\omega_{i_1\cdots i_p} dx^{i_1}\wedge\cdots\wedge dx^{i_p},$$

where the components are completely antisymmetric. There are $\binom np$ independent components, so $\Lambda^pT_p^*M=0$ for $p>n$. Functions are $0$-forms, covector fields are $1$-forms, and top forms have degree $n$.

The factorial matches the site’s weighted antisymmetrization convention. One may instead sum only over strictly increasing indices and omit it. Mixing the two conventions double-counts components. The exterior-algebra construction and dimension count are developed in Guillemin and Haine 2019, §§1.4–1.6, pp. 13–25, PDF.

Wedge products and graded signs

For $\alpha\in\Lambda^pV^*$ and $\beta\in\Lambda^qV^*$, the wedge product is the alternating part of $\alpha\otimes\beta$, normalized so that coordinate basis forms concatenate in the expected way. It is associative, bilinear, and graded commutative:

$$\alpha\wedge\beta=(-1)^{pq}\beta\wedge\alpha.$$

Thus two one-forms anticommute, while a two-form commutes with a one-form. For one-forms $\alpha$ and $\beta$,

$$(\alpha\wedge\beta)(X,Y) =\alpha(X)\beta(Y)-\alpha(Y)\beta(X).$$

In $\mathbb R^3$, if $\alpha=a_i dx^i$ and $\beta=b_i dx^i$, the coefficients of $\alpha\wedge\beta$ are the antisymmetric combinations $a_ib_j-a_jb_i$. This statement resembles a cross product but requires neither a metric nor an orientation. A cross product would require both.

For a vector field $X$, the interior product lowers degree:

$$(\iota_X\omega)(X_2,\ldots,X_p) =\omega(X,X_2,\ldots,X_p).$$

It obeys the graded product rule

$$\iota_X(\alpha\wedge\beta) =(\iota_X\alpha)\wedge\beta +(-1)^p\alpha\wedge\iota_X\beta.$$

The sign follows by moving $X$ past the $p$ slots of $\alpha$.

Pullback and the determinant mechanism

Every smooth map $F:M\to N$ pulls a form on $N$ back to $M$:

$$(F^*\omega)_p(v_1,\ldots,v_p) =\omega_{F(p)}(dF_pv_1,\ldots,dF_pv_p).$$

Pullback preserves degree and wedge products. If $F$ is a coordinate change on an $n$-manifold, then

$$F^*(dy^1\wedge\cdots\wedge dy^n) =\det\!\left(\frac{\partial y}{\partial x}\right) dx^1\wedge\cdots\wedge dx^n.$$

The determinant is precisely the signed volume-scaling factor in the change-of-variables theorem. A top form therefore carries both an infinitesimal magnitude and an orientation. Integration will be coordinate independent when chart orientations and Jacobian signs are handled consistently. Tong 2019, §2.4, pp. 76–90, PDF gives a physics-facing account of the same relationship.

Metric-free scope and common distinctions

Forms exist on every smooth manifold. A metric is not needed for $\wedge$, pullback, or $\iota_X$. The Hodge star, a metric volume form, a norm $\lVert\omega\rVert$, and the codifferential do require a metric and, for the star as usually written, an orientation. Keeping these layers separate prevents an accidental dependence on Lorentzian signature in statements that are purely differential-topological.

Alternating does not mean “small under exchange”; it means an exact sign rule. Nor is every antisymmetric component array automatically a global form: its components on overlaps must transform by the covariant tensor law. Finally, a top form can be integrated only after an orientation choice, unless it is replaced by a density or twisted form.

Dimension and sign checks

The dimension formula gives fast checks on component expressions. In four dimensions, a two-form has six independent components, a three-form has four, and a four-form has one. If a calculation claims more after antisymmetrization, indices have been counted more than once. If a five-form appears, it vanishes identically rather than representing a new field.

Graded commutativity is equally diagnostic. Every odd-degree form wedges with itself to zero, but an even-degree form need not. On four-dimensional coordinate space, take

$$\omega=dx^1\wedge dx^2+dx^3\wedge dx^4.$$

The square of either summand is zero because a basis one-form is repeated, but the two cross terms have the same sign:

$$\omega\wedge\omega =2\,dx^1\wedge dx^2\wedge dx^3\wedge dx^4\ne0.$$

This also proves that $\omega$ cannot be a single wedge $\alpha\wedge\beta$ of two one-forms: such a wedge has square zero. Alternating forms can describe a sum of oriented area measurements rather than one simple area element.

Naturality provides a third check. Pulling back a wedge product must agree with wedging the individual pullbacks. If a coordinate calculation violates this identity, the error is usually an omitted Jacobian factor or a reversed basis-form order. These algebraic tests are particularly valuable because they do not depend on any metric signature.

Interior multiplication has its own check: applying it twice with the same vector gives zero, since the form then receives two identical arguments. With two different vectors, the two insertion operators anticommute. These facts follow directly from alternation and are often faster to verify than a component expansion. Together with the wedge-product rule, they prepare the Cartan identities used later for Lie derivatives and connection forms.

Exercises

On $\mathbb R^4$, let $\alpha=dx+2dy$ and $\beta=dy+dz$. Compute $\alpha\wedge\beta$, evaluate it on $X=\partial_x+\partial_z$ and $Y=\partial_y$, and verify the result by the determinant formula for two one-forms.

Solution

Expanding and using $dy\wedge dy=0$ gives

$$\alpha\wedge\beta =dx\wedge dy+dx\wedge dz+2dy\wedge dz.$$

Now $\alpha(X)=1$, $\beta(Y)=1$, $\alpha(Y)=2$, and $\beta(X)=1$. Hence

$$(\alpha\wedge\beta)(X,Y) =\alpha(X)\beta(Y)-\alpha(Y)\beta(X)=1-2=-1.$$

Evaluating the expanded two-form gives the same answer: $dx\wedge dy$ contributes $1$, $dx\wedge dz$ contributes $0$, and $2dy\wedge dz$ contributes $-2$.

References

• Guillemin, Victor, and Peter J. Haine. Differential Forms. World Scientific, 2019, §§1.4–1.6, pp. 13–25, PDF.
• Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §2.4, pp. 76–90, PDF.